Numerical Techniques for Shallow Water Waves

Version 1.2

Numerical Techniques for the Shallow Water Equations

��

The University of Reading, Department of Mathematics, P.O.Box 220,

Whiteknights, Reading, Berkshire, RG6 6AX, UK

E-mail: [email protected]

Numerical Analysis Report 2/99

Abstract

In this report we will discuss some numerical techniques for approximating the

Shallow Water equations. In particular we will discuss finite difference schemes,

adaptations of Roe’s approximate Riemann solver and the Q-Schemes of

Bermudez & Vazquez with the objective of accurately approximating the solution of

the Shallow Water equations. We consider four different test problems for the

Shallow Water equations with each test problem making the source term more

significant, i.e. the variation of the Riverbed becomes more pronounced, so that

the different approaches discussed in this report can be rigorously tested. A

comparison of the different approaches discussed in this report will also be made

so that we may determine which approach produced the most accurate numerical

results overall.

The work contained in this report has been carried out as part of the Oxford / Reading

Institute for Computational Fluid Dynamics and was funded by the Engineering and Physical

Science Research Council and HR Wallingford under a CASE award.

1

1 Introduction

Throughout this report, we will be discussing some numerical techniques for

approximating the Saint-Venant equations, i.e.

( ) ( )wRwFw

x,xt

=∂

∂+∂∂

(1.1)

where

( ) ⎥⎦

⎤⎢⎣

⎡=

uh

htx,w , ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡+= ghhu

uh22

2

1wF and ( ) ( )⎥⎦⎤

⎢⎣

⎡′

=xHgh

x0

, wR .

Here, h(x,t) and u(x,t) represent the water depth and the fluid velocity respectively and

H(x) is the bed depth from a fixed reference level, see Figure 1-1.

Figure 1-1: Shallow Water Variables.

D

L

x

H(x)

h(x,t)

Riverbed

River

B(x)

2

Notice that (1.1) has a source term present, ( )wR ,x , which can cause difficulties in

accurately approximating (1.1), especially if the source term is stiff (see Hudson[7]

and LeVeque & Yee[9]). Sometimes, it is convenient to re-write the source term in

terms of the bed height

( ) ( )xBDxH −= ⇒ ( ) ( )xBxH ′−=′ ,

hence,

( ) ( ) ( )⎥⎦⎤

⎢⎣

⎡′−

=⎥⎦

⎤⎢⎣

⎡′

=xBghxHgh

x00

,wR .

In Chapter 2, we will discuss four test problems for the Shallow Water Equations

(1.1) where the source term becomes more significant for each test problem, i.e. the

variation of the riverbed becomes more pronounced. The first test problem will have

no source term present and the second and third test problem will have a source term

present, with the third test problem having a source term more significant than the

second test problem. The final test problem will also have a source term which is the

most significant out of all of the four test problems and thus is the most difficult to

approximate accurately.

The four test problems will allow us to rigorously test the different numerical

approaches for approximating the Shallow Water Equations (1.1), which are discussed

in Chapter 3, and determine if the approaches are accurate when the source term

becomes difficult to approximate accurately. The numerical approaches we will

discuss in Chapter 3 are the finite difference approach, adaptations of Roe’s

approximate Riemann solver and the Q-Schemes of Bermudez & Vazquez[1]. Some

of these approaches require the Jacobian matrix of ( )wF

⎥⎦

⎤⎢⎣

⎡−

=∂∂=

uugh 2

10)(

2wF

wA ,

3

which has eigenvalues

ghu +=λ1 and ghu −=λ2

and eigenvectors

⎥⎦

⎤⎢⎣

⎡+

=ghu

11e and ⎥

⎦

⎤⎢⎣

⎡−

=ghu

12e .

All of the numerical approaches discussed will derive numerical schemes that are

either first order, second order or flux-limited second order schemes (see

LeVeque[10], Kroner[8] and Sweby[14]). In Chapter 4 the different numerical

approaches discussed in Chapter 3 will be compared by using the four test problems

so that we may determine which approach produced the most accurate numerical

results overall.

4

2 Test Problems

In this chapter, we will discuss four test problems, for the Shallow Water Equations

(1.1). The first test problem is the dam-break problem and was discussed by

Glaister[3] and Stoker[13]. This problem is one of the most basic problems for (1.1)

since no source term is present. The second test problem contains a source term and

represents a dam breaking on a variable depth riverbed. The third problem was

discussed by LeVeque[11] and represents a square pulse, which breaks up into two

waves travelling in opposite directions, on a variable depth riverbed. The final test

problem was discussed by Bermudez & Vazquez[1] and represents a tidal wave

propagating on a variable depth riverbed. For each test problem, notice how the

source term becomes more significant, i.e. the variation of the bed depth becomes

more pronounced.

2.1 Problem A - The Dam-Break Problem

In this test problem, (1.1) has no source term present, i.e.

( )0=

∂∂+

∂∂

xt

wFw. (2.1)

This is due to the riverbed being of constant depth resulting in ( ) 0=′ xH ∀x. We also

have initial conditions

u(x,0) = 0 and ( )⎪⎪⎩

⎪⎪⎨

⎧

≤<φ

≤≤=

12

1 if

2

10 if 1

0,

0 x

xxh ,

5

which are illustrated in Figure 2-1. In Figure 2-1, the discontinuity at x = 0.5

represents a barrier, which separates the two initial river heights and is removed at t =

0. Walls are present at x = 0 and at x = 1 resulting in reflection at both boundaries.

Note that, for this test problem, if 2.71

0

>φ

then both eigenvalues of )(wA are of the

same sign and the downstream flow is supercritical. If 2.71

0

<φ

then the eigenvalues

of )(wA are of opposite sign and the downstream flow is subcritical. If the

downstream flow is supercritical then difficulties can arise in accurately numerically

approximating the wave speed of the discontinuity at x = 0. In Figure 2-1,

2.725.0

11

0

<==φ

, hence the downstream flow is subcritical.

We may obtain an exact solution of the dam-break problem by using the analysis of

Stoker[13]. The following exact solution is illustrated in Figure 2-2 and Figure 2-3

and is only valid for φ0 = 0.5 and φ1 = 1,

( )( ) ( )

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+>

+≤<+−

+−≤≤−+−

−<

=

2

1S if 0

2

1S

2

1 if

2

1

2

1 if 212

3

12

1 if 0

,

222

22

tx

txtcuu

tcuxgt gtxt

gtx

txu

and

( )( )

( )

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

+>

+≤<+−⎟⎟

⎠

⎞⎜⎜

⎝

⎛−+

+−≤≤−⎟⎠⎞⎜

⎝⎛ −−

−<

=

21

S if 21

21

S21

if 116

141

21

21

if 2

122

91

21

if 1

,

22

2

22

2

tx

tx tcug

S

tcuxgt t

xg

g

gtx

txh

6

where

⎟⎟⎠

⎞⎜⎜⎝

⎛++−=

gS

S

gSu

16118

2

2 ,

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 1161

4

2

2gSg

c

and the wave speed of the discontinuity created at x = 0 is

S = 2.957918120187525.

For a more in depth analysis on how the value of S was obtained see Glaister[3] and

Stoker[13].

2.2 Problem B - The Dam-Break Problem on a Variable

Depth Riverbed

This test problem is similar to Problem A but the riverbed is no longer of constant

depth, resulting in ( ) 0≠′ xH for some values of x, which means that a source term is

present, i.e.

( ) ( )wRwFw

,xxt

=∂

∂+∂∂

.

For this test problem, the riverbed is defined as

⎪⎩

⎪⎨

⎧≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −π

=

otherwise 0

5

3

5

2 if 1

2

110cos

8

1)(

xxxB

and we have initial conditions

u(x,0) = 0 and ( )( )

( )⎪⎪⎩

⎪⎪⎨

⎧

≤<−φ

≤≤−=

12

1 if

2

10 if 1

0,

0 x xB

xxBxh ,

7

which is illustrated in Figure 2-4. In Figure 2-4, the discontinuity at x = 0.5 represents

a barrier, which separates the two initial river heights and is removed at t = 0. Walls

are again present at x = 0 and at x = 1 giving reflection at these boundaries. Since a

source term is now present, difficulties can arise when numerically approximating this

problem, as we will see later.

2.3 Problem C - A Problem Discussed by LeVeque[11] In this test problem, the riverbed is again of variable depth, resulting in ( ) 0≠′ xH for

some values of x, which means that a source term is present, i.e.

( ) ( )wRwFw

,xxt

=∂

∂+∂∂

.

For this test problem, the riverbed is defined as

⎪⎩

⎪⎨

⎧≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞⎜

⎝⎛

⎟⎠⎞

⎜⎝⎛ −π

=

otherwise 0

5

3

5

2 if 1

2

110cos

4

1)(

xxxB

and we have initial conditions

u(x,0) = 0 and ( )( )( )

( )⎪⎩

⎪⎨⎧

>−≤≤−

<−=

0.2 if 1

0.20.1 if 1.2

1.0 if 1

0,

xxB

xxB

xxB

xh ,

which is illustrated in Figure 2-5. This problem represents an initial pulse, which

breaks up into two waves moving in opposite directions. The right-going square-

wave pulse passes the hump in the riverbed and becomes partially reflected, causing a

disturbance behind the hump. In this problem, there are no walls present at x = 0 and

x = 1 and reflection does not occur.

8

2.4 Problem D - Tidal Wave Propagation on a Variable

Depth Bed

This test problem was discussed by Bermudez and Vazquez[1] and also has a source

term present, i.e.

( ) ( )wRwFw

,xxt

=∂

∂+∂∂

.

We have a variable depth given by

⎟⎠⎞⎜

⎝⎛

⎟⎠⎞⎜

⎝⎛ +π+−=

2

14sin10

405.50)(

L

x

L

xxH ,

with initial conditions

u(x,0) = 0 and h(x,0) = H(x),

which are illustrated in Figure 2-6, and boundary conditions

⎪⎩

⎪⎨

⎧

>

≤⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎠⎞⎜

⎝⎛ −+

=43200 if 5.60

43200 if 21

864004

sin45.64),0(

t

tt

thπ

,

representing an incoming wave which is illustrated in Figure 2-7, and u(L,t) = 0. For

this test problem, the reference depth D is taken as D = 60.5.

This problem represents a tidal wave propagating on a variable depth riverbed. Here,

h(0,t) represents a tidal wave of 4m amplitude and in Figure 2-7, we can see that at t =

21,600s, the tidal wave has reached it’s full height of 8m at inflow whereas at t =

43,200s, the tidal wave has disappeared. Also, since the waves propagate at speed

gh , the tidal wave should only reach as far as 216,000m at t = 10,800s, when L =

648,000m.

Throughout this chapter, we have discussed four different test problems for the

Shallow Water Equations. In the next chapter, we will discuss some numerical

9

techniques for approximating the Shallow Water Equations so that we can apply the

different numerical techniques to the four test problems.

Initial Conditions for Problem A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

h(x,

0)

Figure 2-1: Initial condition h(x,0) for Problem A

10

Exact Solution of Problem A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

h(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 2-2: Illustration of the exact solution of Problem A

E xact S olution of P roblem A.

0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

0 .9

1

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

x

u(x,

t)

0 0 .01 0 .02 0 .03 0 .04 0 .05 0 .06 0 .07 0 .08 0 .09 0 .1 Figure 2-3: Illustration of the exact solution of Problem A

11

Initial Conditions for Problem B

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

B(x) h(x,0)+B(x)

Figure 2-4: Initial condition h(x,0) and B(x) for Problem B.

Initial Conditions for Problem C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

B(x) h(x,0)+B(x)

Figure 2-5: Initial condition h(x,0) and B(x) for Problem C.

12

Initial Conditions for Problem D

0

10

20

30

40

50

60

70

0 2000 4000 6000 8000 10000 12000 14000

x

RiverBed River

Figure 2-6: Initial condition h(x,0) and B(x) for Problem D.

Boundary Condition h(0,t) for Problem D.

0

1

2

3

4

5

6

7

8

9

10

0 10800 21600 32400 43200 54000 64800

t

60.5

- h

(0,t)

Figure 2-7: Boundary condition h(0,t) for Problem D.

13

3 Numerical Schemes

There are a variety of numerical techniques for approximating (1.1), e.g. finite

element methods, finite volume methods, etc. In this report, we will discuss the finite

difference approach (see LeVeque[10] and Kroner[8]), adaptations of Roe’s

approximate Riemann solver (see Glaister[4], Hubbard[6] and Roe[12]) and the Q-

Schemes of Bermudez & Vazquez[1].

These approaches will be used to derive first order and second order numerical

schemes where if a numerical scheme is first order then the scheme is dissipative and

if a numerical scheme is second order then the scheme is dispersive (see Figure 3-1).

Dissipation occurs when the travelling wave’s amplitude decreases resulting in the

numerical solution being smeared. Dispersion occurs when waves travel at different

wave speeds and results in oscillations being present in the numerical results. Both

dissipation and dispersion can cause very significant errors in the numerical results,

see Figure 3-1, and can sometimes give completely inaccurate numerical results.

One way to minimise dissipation and dispersion is to use a numerical method which

satisfies the Total Variational Diminishing property (see Sweby[14] and Harten[5]).

Flux-limiter methods satisfy the TVD property and switch between a second order

approximation when the region is smooth and a first order approximation when near a

discontinuity. Flux-limiter methods will also be applied to the different numerical

approaches so that oscillations present in the numerical solution can be minimised.

14

Numerical Results of an Advection Test Problem using First and Second Order Finite Difference Schemes with the Exact Solution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.5)

Exact Solution First Order Scheme with Dissipation present Second Order Scheme with Dispersion Present

Figure 3-1: Illustration of Dispersion and Dissipation

Figure 3-2: The Mesh

x = x0

i = 0 x = x1 i = I

x x1 x2 x3 …. xI-3 xI-2 xI-1 xi-1 xi xi+1 ……

n = 0 t0 = 0

u(x,0)

tn-1

t1

t2

…

Δt tn

tn+1

uni un

i 1+ uni 1−

uni

11++ un

i1+ un

i11+−

uni

11−− un

i1− un

i11−+

..….

Δx

t

n = N t = tN

tN-1

tn+1 – tn = Δt xi+1 – xi = Δx

tn = nΔt xi = iΔx

tN-2

15

3.1 Mesh & Boundary Conditions Before we discuss the various numerical approaches for approximating the Shallow

Water Equations (1.1), we must define the mesh and then look at the numerical

boundary conditions required to implement the numerical approaches correctly.

In this report, we will use a fixed mesh over the finite region xxx Io ≤≤ and tt N≤≤0 ,

which is illustrated in Figure 3-2. Here, the numerical solution is denoted by

( )tnxiuuni ΔΔ≈ , where xxx ii 1−−=Δ and ttt nn 1−−=Δ for all i and n.

Numerical boundary conditions are required at x = x0 and x = xI and are derived

separately for each test problem. For

i) Problem A and Problem B, walls are present at the upstream boundary, x

= 0, and the downstream boundary, x = 1, so we will need to reflect the velocity at the

two boundaries

uu ni

ni −=− and uu n

iIn

iI −+ −=

where i = 1,2. For the water depth, we will assume that the depth is constant at the

boundaries

hh nni 0=− and hh n

In

iI =+ where i = 1, 2.

ii) Problem C we will assume that the velocity and water depth are constant

hh nni 0=− , hh n

In

iI =+ , uu nni 0=− and uu n

In

iI =+

where i = 1, 2.

iii) Problem D we use the analytical boundary conditions to obtain

( )thhni ,0=− and 0=+un

iI

where i = 0, 1, 2. For the second boundary condition, we will use

hh iIn

iI0++ = and uu nn

i 0=−

where i = 1, 2.

16

3.2 Finite Difference Method One approach widely used to numerically approximate (1.1) is the finite difference

method. This method involves replacing the derivatives of (1.1) with finite difference

approximations, e.g.

tt

ni

ni

Δ−=

∂∂ + www 1

which is a forward difference approximation in time, to obtain a finite difference

scheme. Great care must be taken when using finite differences to construct a finite

difference scheme as we need to ensure that the scheme is conservative. A finite

difference scheme that is not conservative may propagate discontinuities at the wrong

wave speed, if at all, giving inaccurate numerical results. To ensure we obtain a

conservative scheme, we only construct finite difference schemes of the form

[ ]FFww *2/1

*2/1

1−+

+ −ΔΔ−= ii

ni

ni

x

t, (3.1)

where F* is called the numerical flux function. Notice that (3.1) approximates the

homogeneous problem (2.1). Obtaining a conservative scheme when a source term is

present can be very difficult, especially when the source term is stiff. However, there

are a variety of numerical techniques which approximate the source term and they

generally fall under two categories: a pointwise approach, where we add a source term

approximation to (3.1), i.e.

[ ] RFFww tx

t niii

ni

ni Δ+−

ΔΔ−= −+

+ *2/1

*2/1

1 (3.2)

or a physical approach, which involves taking an average value of the source term and

upwinding (see Sweby[15] for more details), i.e.

[ ] RFFww tx

tiii

ni

ni Δ+−

ΔΔ−= −+

+ **2/1

*2/1

1 . (3.3)

17

We will now look at specific schemes which numerically approximate the Shallow

Water Equations7.

3.2.1 First Order Lax- Friedrichs Approach We can use the Lax-Friedrichs scheme with a source term approximation ‘added’ to

approximate (1.1), i.e.

( ) RFFww

w ss n

ini

ni

ni

nin

i +−−+= −+−++

11111

22 (S-1)

where

x

ts

ΔΔ= , ⎥

⎦

⎤⎢⎣

⎡=

uh

hni

ni

nin

iw , ( ) ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=hguh

uhni

ni

ni

ni

ni

ni 2 2

2

1F and ( )⎥⎦⎤

⎢⎣

⎡−−

=−BBhg ii

ni

ni

1

0R .

Notice that this scheme is in a similar form of (3.2) and that the Jacobian matrix of

F(w) associated with the system (1.1) does not need to be approximated. However,

the Lax-Friedrichs scheme can suffer from oscillations but sometimes these

oscillations can be minimised by using sufficiently small step-sizes.

3.2.2 Explicit MacCormack Approach We can also use an approach discussed by LeVeque & Yee[9] to approximate (1.1),

which involves modifying the explicit MacCormack scheme, i.e.

( ) ( ) RFFwww )1()1(1

)1()1(1

222

1iiii

ni

ni

ss +−−+= −+ (S-2a)

where ( ) RFFww ss n

ini

ni

nii +−−= +1

)1( ,

x

ts

ΔΔ= , ⎥

⎦

⎤⎢⎣

⎡=

uh

hni

ni

nin

iw , ( ) ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=hguh

uhni

ni

ni

ni

ni

ni 2 2

2

1F and ( )⎥⎦⎤

⎢⎣

⎡−−

=−BBhg ii

ni

ni

1

0R .

Unfortunately, this scheme suffers from dispersion and results in oscillations being

present in the numerical approximation. However, LeVeque & Yee[9] eliminated

18

these oscillations by adapting (S-2a) so that the scheme satisfies the TVD property

and obtained

( )ϕϕ 2/12/12/1 2/1)2(1

−−+ ++ −+= iii ii

ni XXww (S-2b)

where X represents a matrix containing the right eigenvectors e, w )2(

i is the numerical

approximation derived from scheme (S-2a), i.e.

( ) ( ) RFFwww )1()1(1

)1()1()2(

222

1iiii

nii

ss +−−+= − ,

and

⎥⎦

⎤⎢⎣

⎡φφ

=+

++ 2

2/1

12/1

2/1

i

iiϕ

where

( )( )( )Qvvki

ki

ki

ki

ki 2/12/12/1

22/12/1 2

1+++++ −α−=φ ,

( )ααα= ++−+ki

ki

ki

kiQ 2/32/12/12/1 ,,modmin ,

( ) ( ) ( ) ( ) ( )⎩⎨⎧ ===

=otherwise 0

sgnsgnsgn if ,,min,,modmin

cbadcbadcba ,

( )wwX ni

nii

i

i −=⎥⎦

⎤⎢⎣

⎡αα

+−+

+

+1

12/12

2/1

12/1 and λΔ

Δ= ++ki

ki

x

tv 2/12/1 .

Here, k represents the kth component of the vector and λ, e and α represents the

eigenvalues, eigenvectors and wave strengths associated with the system (1.1)

respectively, which can be obtained by the decomposition

wAeF Δ=∑ λα=Δ=

~~~~2

1k

kkk .

Variables which have ˜ represent the Roe average and A~

is the Jacobian matrix

evaluated at Roe’s average state. We will discuss the method of decomposition in

Section 3.3.

19

3.2.3 Semi-Implicit MacCormack Approach LeVeque & Yee[9] also obtained a semi-implicit approach to approximate (1.1),

which involves modifying the explicit MacCormack scheme so that

( )ϕϕ 2/12/12/1 2/1)2(1

−−+ ++ −+= iii ii

ni XXww (3.4)

where

( ) ( ) RwR

IFFwR

Iwww )1(

1

)1(1

)1(

1

)1()2(

22222

1iiii

nii

ssss⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−+−⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−−+=

−

−

−

,

( ) RwR

IFFwR

Iww ni

ni

ni

nii

ss

ss ⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−+−⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−−=

−

+

−

22

1

1

1

)1(

and

( ) ⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦⎤

⎢⎣⎡∂∂

− 0

00

1BBg ii

n

iwR

which implies that

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−

−

−

12

01

2 1

1

BBgss

iiwR

I .

Also, notice that for the Shallow Water Equations

( ) ( ) ( ) RRwR n

iii

niii

niii

ni

BBhgBBhgBBgss =⎥

⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡−−⎥

⎥⎦

⎤

⎢⎢⎣

⎡−−=⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−

−−−

−

111

100

12

01

2I .

(3.5)

Hence, we can re-write (3.4) as

( )ϕϕ 2/12/12/1 2/1)2(1

−−+ ++ −+= iii ii

ni XXww (S-2c)

where

( ) ( ) RFFwR

Iwww )1()1(1

)1(

1

)1()2(

2222

1iiii

nii

sss +−⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−−+= −

−

and

( ) RFFwR

Iww ni

ni

ni

nii s

ss +−⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−−= +

−

1

1

)1(

2.

20

3.3 Approximate Riemann Solvers Roe [12] derived an approach which approximates systems of conservation laws by

using a piecewise constant approximation

( )⎪⎪⎩

⎪⎪⎨

⎧

Δ+<<Δ−

Δ+<<Δ−=

22 if

22 if

,x

xxx

x

xxx

xx

tx

RRR

LLL

n

w

ww ,

where wL and wR represent the piecewise constant states at tn, and determining the

exact solution of a linearised Riemann problem which is related to (2.1)

( ) 0,~ =

∂∂+

∂∂

xtRL

wwwA

w, (3.6)

where ( )wF

wwA∂∂≈RL ,

~ is the linearised Jacobian matrix and ˜ is called the Roe

average. The eigenvalues and eigenvectors of A~

are λ~ and e~ respectively and are

determined from the decomposition

wAeF Δ=∑ λα=Δ=

~~~~1

k

M

kkk

where www LR −=Δ , M is the number of waves present in the system and α~

represents the wave strengths, i.e. wkk Δ=α~ . Once the eigenvalues, eigenvectors and

wave strengths associated with the linearised Riemann problem have been obtained,

Roe’s scheme [12] can be used

( )FFww *2/1

*2/1

1−+

+ −−= iini

ni s , (3.7)

where

( ) ( )( ) ⎟⎠⎞⎜⎝

⎛ ∑ −φ−λα−+== +

++

M

kkkkkk

i

ni

nii v

1 2/11

*2/1

~11~~2

1

2

1eFFF ,

λΔΔ= ~

kkx

tv ,

( )( )αα=θ

±

±

~

~

2/1

2/1

k i

k Ik , ( )( )viI k i 2/1sgn ±−=

and φk can be any of the flux-limiters listed in Table 3-1.

21

Name of Flux-limiter φ(θ) Minmod φ(θ) = max(0,min(1,θ))

Roe’s Superbee φ(θ) = max(0,min(2θ,1),min(θ,2))

van Leer ( )θ+θ+θ

=θφ1

van Albada ( )θ+θ+θ=θφ2

2

1

Table 3-1: Some second order flux-limiters.

3.3.1 Roe’s Scheme with Source Term Added

As presented so far, Roe’s scheme approximates homogeneous systems of

conservation laws. We now extend (3.7) to numerically approximate the Shallow

Water Equations, which has a source term present.

One approach is to add a source term approximation to (3.7)

( ) RFFww ss niii

ni

ni +−−= −++ *

2/1*

2/11 (S-3a)

where

( ) ( )( ) ⎟⎠⎞⎜⎝

⎛ ∑ −φ−λα−+== +

++

M

kkkkkk

i

ni

nii v

1 2/11

*2/1

~11~~2

1

2

1eFFF ,

λΔΔ= ~

kkx

tv ,

( )( )αα=θ

±

±

~

~

2/1

2/1

k i

k Ik , ( )( )viI k i 2/1sgn ±−= .

and

( )⎥⎦⎤

⎢⎣

⎡−−

=−BBhg ii

ni

ni

1

0R .

We now need to obtain the eigenvalues, eigenvectors and wave strengths associated

with the linearised Riemann problem from the decomposition. For the Shallow Water

Equations, we may obtain eigenvalues, eigenvectors and wave strengths (see

Glaister[3] and Hubbard[6])

cu ~~~1 +=λ , cu ~~~

2 −=λ ,

22

⎥⎦

⎤⎢⎣

⎡+

=cu ~~

1~1e , ⎥

⎦

⎤⎢⎣

⎡−

=cu ~~

1~2e ,

( )( )huhuc

h Δ−Δ+Δ=α ~~2

1

2

1~1

and

( )( )huhuc

h Δ−Δ−Δ=α ~~2

1

2

1~2 ,

where

hh

uhuhuLL

LLRR

++=~ and

2~ hhgc LR+= .

A semi-implicit approach of (S-3a) can be obtained by approximating the source term

at tn+1 instead of at tn, i.e.

( ) RFFww ss niii

ni

ni

1*2/1

*2/1

1 +−+

+ +−−= . (3.8)

Now, by using Taylor’s theorem, we may obtain

( ) ⎥⎦⎤

⎢⎣⎡∂∂−+≈ ++

wR

wwRR ni

ni

ni

ni

11

and by substituting into (3.8) and using (3.5), we may obtain

( ) RFFwR

Iww niii

ni

ni s

ss +−⎥⎦

⎤⎢⎣⎡

⎥⎦⎤

⎢⎣⎡∂∂−−= −+

−+ *

2/1*

2/1

1

1

2. (S-3b)

where

( ) ( )( ) ⎟⎠⎞⎜⎝

⎛ ∑ −φ−λα−+== +

++

M

kkkkkk

i

ni

nii v

1 2/11

*2/1

~11~~2

1

2

1eFFF ,

λΔΔ= ~

kkx

tv ,

( )( )αα=θ

±

±

~

~

2/1

2/1

k i

k Ik , ( )( )viI k i 2/1sgn ±−=

and

( )⎥⎦⎤

⎢⎣

⎡−−

=−BBhg ii

ni

ni

1

0R .

23

3.3.2 Roe’s Scheme with Source Term Decomposed

Instead of adding a source term approximation to Roe’s scheme [12], we can use an

approach discussed by Glaister[3]. This method approximates the source term by

decomposing the source term in a similar way that we did for the flux terms, i.e.

Re~~~1

1=∑β

Δ =k

M

kkx

.

Here, β~k are the coefficients of the decomposition of the source term onto the

eigenvectors of the characteristic decomposition (see Glaister[3] and Hubbard[6]).

For the Shallow Water Equations, we may obtain

2

~~1

HcΔ=β and 2

~~2

HcΔ−=β . (3.9)

Alternatively, we can use

2

~~1

BcΔ−=β and 2

~~2

BcΔ=β

for the Shallow Water equations. Once the values of β~k

have been obtained, we can

approximate the source term by using

( )RRR +−

−+ +

Δ= 2/12/1

* 1iii

x

where

( )( )⎟⎠⎞⎜⎝

⎛ λ±∑β== +

±+

~sgn1~~2

1 2

1 2/12/1 k

kkk

ii eR ,

which is a first order approximation of the source term. Hence, Roe’s first order

scheme with the source term decomposed may be written

( ) ( )RRFFww +−

−+−+

+ ++−−= 2/12/1*

2/1*

2/11

iiiini

ni ss (S-3c)

where

( ) ⎟⎠⎞⎜⎝

⎛ ∑ λα−+== +

++

2

1 2/11

*2/1

~~~2

1

2

1k

kkki

ni

nii eFFF

and

24

( )( )⎟⎠⎞⎜⎝

⎛ λ±∑β== +

±+

~sgn1~~2

1 2

1 2/12/1 k

kkk

ii eR .

We can also obtain a second order accurate approximation of the source term and thus

obtain Roe’s second order scheme with source term decomposed

( ) ( )RRFFww +−

−+−+

+ ++−−= 2/12/1*

2/1*

2/11

iiiini

ni ss (S-3d)

where

( ) ( ) ⎟⎠⎞⎜⎝

⎛ ∑ λα−+== +

++

2

1

2

2/11

*2/1

~~~22

1k

kkki

ni

nii

seFFF

and

( )( )⎥⎦⎤

⎢⎣⎡ λ±∑ λβ=

= +

±+

~sgn1~~~2

2

1 2/12/1 k

kkkk

ii

seR .

In addition, we can also use an approach proposed by Hubbard[6] which involves

applying TVD to the source term approximation as well as the conservation law

approximation, i.e.

( ) ( )RRFFww +−

−+−+

+ ++−−= 2/12/1*

2/1*

2/11

iiiini

ni ss (S-4)

where

( ) ( )( ) ⎟⎠⎞⎜⎝

⎛ ∑ −φ−λα−+== +

++

M

kkkkkk

i

ni

nii v

1 2/11

*2/1

~11~~2

1

2

1eFFF ,

( ) ( )( )( )⎥⎦⎤

⎢⎣⎡ −φ−λ±∑β=

= +

±+ vkkk

kkk

ii 11~sgn1~~

2

1 2

1 2/12/1 eR ,

λΔΔ= ~

kkx

tv ,

( )( )αα=θ

±

±

~

~

2/1

2/1

k i

k Ik , ( )( )viI k i 2/1sgn ±−=

and φk can be any of the flux-limiter functions listed in Table 3-1.

3.3.3 The C – Property of Bermudez & Vazquez

The advantage of decomposing the source term as well as the conservation law is that

we obtain a numerical scheme that satisfies the approximate C-property. Bermudez &

Vazquez[1] state that in order for a numerical scheme to satisfy

25

i) The approximate C-property, the numerical scheme must be at least

second order accurate when applied to the quiescent flow case, i.e. u ≡

0 and h ≡ H.

ii) The exact C-property, the numerical scheme must be exact when

applied to the quiescent flow case, i.e. u ≡ 0 and h ≡ H.

Hubbard’s approach and Roe’s scheme with source term decomposed all satisfy the

exact C-property and should produce very accurate numerical results. However,

Roe’s scheme with source term added does not even satisfy the approximate C-

property and may give misleading results as the source term becomes significant.

3.4 Q-Schemes of Bermudez & Vazquez Bermudez & Vazquez[1] discussed a variety of Q-Schemes, which numerically

approximate (1.1). All of the Q-Schemes discussed were used with the following first

order equation

( ) RFFww ts niii

ni

ni Δ+−−= −++ *

2/1*

2/11 (S-5)

where

( ) ( )( )wwwwQFFF ni

ni

ni

ni

ni

nii −−+= ++++ 111

*2/1 ,

2

1

2

1,

( ) ( )wwRwwRR ni

niiiR

ni

niiiL

ni xxxx 1111 ,,,,,, ++−− += ,

( ) ( ) ( )[ ] ( )wwRwwQwwQIwwR ni

niii

ni

ni

ni

ni

ni

niiiL xxxx ,,,ˆ,,

2

1,,, 111

1111 −−−

−−−− += ,

( ) ( ) ( )[ ] ( )wwRwwQwwQIwwR ni

niii

ni

ni

ni

ni

ni

niiiR xxxx 111

1111 ,,,ˆ,,

2

1,,, +++

−+++ −= ,

( ) ( ) ( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛

Δ−

⎟⎠⎞⎜

⎝⎛ += ++++

xxHxHhhgxx ii

ni

ni

ni

niii 1111

2

0,,,ˆ wwR

26

and Q is a matrix calculated by using a certain Q-Scheme. Bermudez & Vazquez[1]

discussed a variety of Q-Schemes but generally concentrated on the Q-scheme of van

Leer and a Q-Scheme which is equivalent to Roe’s first order scheme (S-3c).

The Q-Scheme of van Leer is

( ) ⎟⎠⎞

⎜⎝⎛ += +

+2

, 11

wwAwwQni

nin

ini ,

where A denotes the Jacobian matrix of )(wF

⎥⎦

⎤⎢⎣

⎡−

=uugh 2

10)(

2wA ,


ghu +=λ1 and ghu −=λ2

and eigenvectors

⎥⎦

⎤⎢⎣

⎡+

=ghu

11e and ⎥

⎦

⎤⎢⎣

⎡−

=ghu

12e .

The Q-Scheme that is equivalent to Roe’s first order scheme is

( ) ( ) ⎥⎦

⎤⎢⎣

⎡−

==+uuc

ni

ni ~2~~

10~,221 wAwwQ ,


cu ~~~1 +=λ , cu ~~~

2 −=λ

and eigenvectors

⎥⎦

⎤⎢⎣

⎡+

=cu ~~

1~1e , ⎥

⎦

⎤⎢⎣

⎡−

=cu ~~

1~2e ,

where

hh

uhuhuLL

LLRR

++=~ and

2~ hhgc LR+= .

For both Q-Schemes, the modulus of matrix A can be obtained by using

27

X�XA 1−=

where Λ represents a matrix whose diagonal elements are the eigenvalues of A,

⎥⎦

⎤⎢⎣

⎡λ

λ=2

1

0

0�

and X denotes a matrix containing the right eigenvectors,

⎥⎦

⎤⎢⎣

⎡λλ

=21

11)(wX .

Hence, for both Q-Schemes we may obtain

( ) ⎥⎦

⎤⎢⎣

⎡λλ−λλλ−λλλ

λ−λλλ−λλλ−λ

=11222112

122121

12

1A ,

and for the source term approximation we may obtain

( ) ( )( )( )

( )( )⎥⎦

⎤⎢⎣

⎡λ−λ+λ−λλλ

λλ−λλλλλ−λ

=−−

−−−−

121212

1221

2/11212 2/1

11211

,,,ˆ,,,ˆ

ii

ni

niiin

iniiiL

xxxx

wwRwwR

and

( ) ( )( )( )

( )( )⎥⎦

⎤⎢⎣

⎡λ+λ−λ−λλλ

λλ−λλ−

λλλ−λ=

++

++++

121212

1221

2/11212 2/1

11211

,,,ˆ,,,ˆ

ii

ni

niiin

iniiiR

xxxx

wwRwwR

where

( ) ( ) ( ) ( )( )xHxHhhx

gxx ii

ni

ni

ni

niii −+

Δ= ++++ 11112 2

,,,ˆ wwR .

Again, notice that (S-5) with either Q-Scheme satisfies the exact C-property.

In this chapter, we have discussed a variety of numerical schemes which approximate

(1.1). In the next chapter, we will apply these schemes to the four test problems

discussed in Chapter 2 to find out which approach is the most accurate.

28

4 Numerical Results

In this chapter, we will apply the schemes discussed in Chapter 3 and listed in Table

4-1 to the four test problems discussed in Chapter 2 to find out which approach

produces the most accurate results. We will not discuss the results of the semi-

implicit approaches as they produced almost identical results to the explicit

approaches. Also, the numerical results of Bermudez & Vazquez’s Q-Schemes will

not be discussed as the two Q-Schemes produced almost identical results to Roe’s first

order scheme with source term decomposed, i.e. (S-3b).

For the first three test problems, step-sizes Δx = 0.001 and Δt = 0.0001 will be used

with a final time of t = 0.1. A comparison will be made at t = 0.1 and numerical

results will also be shown for t = 0.01m, where m = 0 to 10.

For test Problem D, step-sizes Δx = 2800 and Δt = 1 will be used with a bed length of

L = 648,000m and a final time of t = 10,800s. A comparison will be made at t =

10,800s and numerical results will also be shown for t = 1080m, where m = 0 to 10.

Name Of Approach Reference No. Order PaperLax-Friedrichs (S-1) 1 -

MacCormack (S-2b) 2Yee[16],

LeVeque & Yee[9],Roe’s Scheme with Source Term

‘added’(S-3a) 1 / 2

Roe’s Scheme with Source TermDecomposed

(S-3b) 1

Hubbard[6],

Glaister[3],

Roe[12]

Hubbard’s Approach (S-4) 2 Hubbard[6]Table 4-1: Different approaches for numerically approximating (1.1).

29

Only first order and flux-limited second order numerical results will be discussed and

the Minmod flux-limiter will be used with all flux-limited second order approaches.

4.1 First Order Comparison

4.1.1 Numerical Results for Problem A

For this test problem, no source term is present so schemes (S-3a) and (S-3b) are the

same. Now, by using schemes (S-1) and (S-3b) to approximate Problem A and

comparing with the exact solution, we may obtain the numerical results in Figure 4-1a

and Figure 4-1b. Here, we can see that Roe’s scheme is more accurate than the Lax-

Friedrichs approach since the Lax-Friedrichs approach is more dissipative than Roe’s

scheme. Also, the Lax-Friedrichs scheme suffered badly from oscillations if larger

step-sizes were used whereas Roe’s scheme remained accurate but became more

dissipative.

4.1.2 Numerical Results for Problem B

For this test problem, a source term is now present so approaches (S-3a) and (S-3b)

are no longer equivalent. By using schemes (S-1), (S-3a) and (S-3b) to approximate

Problem B, the results in Figure 4-2a to Figure 4-3b were obtained. Here we can see

that Roe’s scheme with source term decomposed has produced the most accurate

results. Roe’s scheme with source term added produced almost identical results to

Roe’s scheme with source term decomposed showing that, for Problem B, adding a

source term approximation to Roe’s scheme produces sufficiently accurate results.

The Lax-Friedrichs approach produced the least accurate results suffering badly from

dissipation and the approach also misplaced the disturbance caused by the riverbed.

30

4.1.3 Numerical Results for Problem C

For this test problem, the source term is becoming more significant, i.e. the variation

in the riverbed is becoming more pronounced, which may cause some schemes to

produce inaccurate results. By using approaches (S-1), (S-3a) and (S-3b) to

approximate Problem C, the results in Figure 4-4a to Figure 4-7b were obtained.

Here, we can see that Roe’s scheme with source term decomposed has produced the

most accurate results but the results are no longer almost identical to Roe’s scheme

with source term added. Adding the source term in this case has produced movement

for 0.3 > x > 0.55 whereas decomposing the source term has produced no movement

for 0.3 > x > 0.55. This is because Roe’s scheme with source term decomposed

satisfies the exact C-property whereas Roe’s scheme with source term added does not.

The Lax-Friedrichs approach has produced the least accurate results due to the

scheme suffering badly from dissipation and the scheme has also produced more

movement than Roe’s scheme with source term added for 0.3 > x > 0.55. Also, from

Figure 4-7b we can see that the Lax-Friedrichs approach has started producing

oscillations at the peak of the pulse even though small step-sizes have been used.

4.1.4 Numerical Results for Problem D

For this test problem, the source term is very difficult to approximate accurately

which may cause some approaches to produce very inaccurate numerical results. By

applying schemes (S-1), (S-3a) and (S-3b) to Problem D, the results in Figure 4-8a to

Figure 4-11b were obtained. Here, we can see that Roe’s scheme with source term

decomposed has produced the most accurate results since it was the only approach not

to produce movement after x = 216,000m at t = 10,800s. This is because the

numerical scheme satisfies the exact C-property whereas the other approaches do not

even satisfy the approximate C-property. Roe’s scheme with source term added was

31

the second most accurate due to the approach producing movement after x = 216,000

at t = 10,800s. The Lax-Friedrichs approach failed to accurately approximate

Problem D at all due to the scheme producing oscillations over the whole domain.

4.1.5 Overall Comparison of First Order Approaches

From the results of this sub-section, we have seen that Roe’s scheme with source term

decomposed has produced very accurate results for all test problems suffering only

slightly from dissipation. Roe’s scheme with source term added produced accurate

numerical results for the first two test problems, but as the source term became more

significant, the numerical scheme started to produce less accurate results. In Problem

D, Roe’s scheme with source term added produced movement after x = 216,000 at t =

10,800s making the scheme very inaccurate. The Lax-Friedrichs approach is accurate

for the most basic test problems but only when sufficiently small step-sizes are used

otherwise the scheme suffers from oscillations. Also, as the source term became

significant the Lax-Friedrichs approach became impractical suffering badly from

oscillations even when small step-sizes were used. Hence, Roe’s scheme with source

term decomposed produced the most accurate results for all test problems in the first

order case since it was the only approach to satisfy the exact C-property.

4.2 Flux-Limited Second Order Comparison

4.2.1 Numerical Results for Problem A

Now, by using approaches (S-2b) and (S-4) to approximate Problem A, the results in

Figure 4-12a and Figure 4-12b were obtained. Here, we can see that Roe’s scheme

and the MacCormack approach have produced almost identical results. Both

approaches have produced extremely accurate numerical results but Roe’s scheme

was the most accurate.

32

4.2.2 Numerical Results for Problem B

By using approaches (S-2), (S-3a) and (S-4) to approximate Problem B, the results in

Figure 4-13a to Figure 4-14b were obtained. Here, we can see that Roe’s scheme with

source term added, Hubbard’s approach and the MacCormack approach have all

produced almost identical results. All three approaches have produced very accurate

results.

4.2.3 Numerical Results for Problem C

By using approaches (S-2), (S-3a) and (S-4) to approximate Problem C, the results in

Figure 4-15a to Figure 4-18b were obtained. Here, we can see that Hubbard’s

approach has produced the most accurate results. The other two approaches have

produced movement in the region 0.3 > x > 0.55 with the MacCormack approach

producing the most amount of movement of the three approaches. If we use a larger

step-size of Δx = 0.01 and Δt = 0.001 with approaches (S-2), (S-3a) and (S-4), we may

obtain the results in Figure 4-19a and Figure 4-19b. Here we can see that Hubbard’s

approach has again produced the most accurate numerical results and has not

produced movement in the region 0.3 > x > 0.55. However, the MacCormack

approach and Roe’s scheme with source term added have both produced very

inaccurate numerical results. Roe’s scheme with source term added has produced

inaccurate numerical results due to the approach producing a considerable amount of

movement in the region 0.3 > x > 0.55 and the approach has slightly smeared the

pulse at x = 0.5. The MacCormack approach has produced less movement in the

region 0.3 > x > 0.55 than Roe’s scheme with source term added but the approach has

smeared the pulse at x = 0.5 considerably more than Roe’s scheme with source term

added. Hence, Hubbard’s approach is still very accurate when the step-size is

increased but the other two approaches are becoming impractical.

33

4.2.4 Numerical Results for Problem D

Now, by using approaches (S-2), (S-3a) and (S-4) to approximate Problem D, the

results in Figure 4-20a to Figure 4-23b were obtained. Here, we can see that

Hubbard’s approach was the only approach that did not produce movement after x =

216,000m at t = 10,800s. Roe’s scheme with source term added and LeVeque &

Yee’s MacCormack approach produced very similar results but both produced

movement after x = 216,000m at t = 10,800s. Hence, Hubbard’s produced the most

accurate results.

4.2.5 Overall Comparison of the Flux- Limited Second

Order Approaches

From the results of this sub-section, we have seen that Hubbard’s approach has

produced the most accurate results for all test problems producing very accurate

results even for Problem D. LeVeque & Yee’s MacCormack approach, Roe’s scheme

with source term added and Hubbard’s approach all produced almost identical results

for the first two test problems. However, as the source term became more significant,

LeVeque & Yee’s MacCormack approach and Roe’s scheme with source term added

produced less accurate results but still produced very similar results due to both

approaches adding a source term approximation on. When the source term became

significant and a larger step-size was used, the MacCormack approach and Roe’s

scheme with source term added both became impractical but Hubbard’s approach still

produced very accurate numerical results.

34

First Order Numerical Results:

Comparison of the Different First Order Results with the Exact Solution using hx = 0.001, ht = 0.0001 and at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

h(x,

0.1)

Exact Lax-Friedrichs Roe's Scheme

Figure 4-1a: Comparison of the first order results for Problem A

Com parison of the Different First Order Results w ith the Exact Solution using hx = 0.001, ht = 0.0001 and at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)

Exact Lax-Friedrichs Roe's Schem e

Figure 4-1b: Comparison of the results for Problem A

35

Roe's Scheme with Source Term Decomposed and hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-2a: Results of Roe’s scheme with source term decomposed for Problem B

Roe's Scheme with Source Term Decomposed and hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-2b: Results of Roe’s scheme with source term decomposed for Problem B

36

Comparison of the Different First Order Approaches at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed Lax-FriedrichsRoe's Scheme with Source Term Added Roe's Scheme with Source Term Decomposed

Figure 4-3a: Comparison of the different first order approaches for Problem B


-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)

Lax-Friedrichs Roe's Scheme with Source Term Added Roe's Scheme with Source Term Decomposed

Figure 4-3b: Comparison of the different first order approaches for Problem B

37

Lax-Friedrichs Approach with hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-4a: Results of Lax-Friedrichs approach for Problem C

Lax-Friedrichs Approach with hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-4b: Results of Lax-Friedrichs approach for Problem C

38

Roe's Scheme w ith Source Term Decomposed and hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-5a: Results of Roe’s scheme with source term decomposed for Problem C

Roe's Scheme w ith Source Term Decomposed and hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-5b: Results of Roe’s scheme with source term decomposed for Problem C

39

Roe's Scheme w ith Source Term Added and hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-6a: Results of Roe’s scheme with source term added for Problem C

Roe's Scheme with Source Term Added and hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-6b: Results of Roe’s scheme with source term added for Problem C

40


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed Lax-FriedrichsRoe's Scheme with Source Term Added Roe's Scheme with Source Term Decomposed

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

0.3 0.35 0.4 0.45 0.5 0.55 0.6

Figure 4-7a: Comparison of the different first order approaches for Problem C


-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)

Lax-Friedrichs Roe's Scheme with Source Term Added Roe's Scheme with Source Term Decomposed

Figure 4-7b: Comparison of the different first order approaches for Problem C

41

Lax-Friedrichs Approach w ith hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Riverbed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-8a: Results of Lax-Friedrichs approach for Problem D

Lax-Friedrichs Approach with hx = 1000, ht = 1 and t = 0 to 10800.

-4

-3

-2

-1

0

1

2

3

4

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-8b: Results of Lax-Friedrichs approach for Problem D

42

Roe's Scheme with Source Term Decomposed and hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Riverbed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-9a: Results of Roe’s scheme with source term decomposed for Problem D

Roe's Scheme w ith Source Term Decomposed and hx = 1000, ht = 1 and t = 0 to 10800.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-9b:Results of Roe’s scheme with source term decomposed for Problem D

43

Roe's Scheme with Source Term Added and hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Seabed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-10a: Results of Roe’s scheme with source term added for Problem D

Roe's Scheme w ith the Source Term Added and hx = 1000, ht = 1 and t = 0 to 10800.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-10b: Results of Roe’s scheme with source term added for Problem D

44

Comparison of the Different First Order Approaches at t = 10800.

0

10

20

30

40

50

60

70

0 216000 432000 648000

x

Riverbed River InitiallyLax-Friedrichs Roe's Scheme with Source Term AddedRoe's Scheme with Source Term Decomposed

Figure 4-11a: Comparison of the different first order approaches for Problem D

Comparison of the Different First Order Approaches at t = 10800.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 216000 432000 648000

x

u(x,

1080

0)

Lax-Friedrichs Source Term Added Source Term Decomposed

-4

-3

-2

-1

0

1

2

3

4

0 216000 432000 648000x

Figure 4-11b: Comparison of the different first order approaches for Problem D

45

Flux-Limited Second Order Numerical Results:

Comparison of the Different Flux-Limited Second Order Approaches with the Exact Solution at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

h(x,

0.1)

Exact Solution MacCormack Approach Roe's Scheme

Figure 4-12a: Comparison of the flux-limited second order results for Problem A

Comparison of the Different Flux-Limited Second Order Approaches with the Exact Solution and at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)

Exact Solution MacCormack Approach Roe's Scheme

Figure 4-12b: Comparison of the flux-limited second order results for Problem A

46

Hubbard's Approach with hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-13a: Results of Hubbard’s approach for Problem B


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-13b: Results of Hubbard’s approach for Problem B

47

Comparison of the Different Flux-Lim ited Second Order Approaches at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed MacCormack Approach Roe's Scheme with Source Term Added Hubbard's Approach

Figure 4-14a: Comparison of the flux-limited second order results for Problem B

Comparison of the Different Flux-Limited Second Order Appraoches at t = 0.1.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)

MacCormack Approach Roe's Scheme with Source Term Added Hubbard's Approach

Figure 4-14b: Comparison of the flux-limited second order results for Problem B

48


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-15a: Results of Hubbard’s approach for Problem C

Hubbard's Approach w ith hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-15b: Results of Hubbard’s approach for Problem C

49


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-16a: Results of Roe’s scheme with source term added for Problem C


-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-16b: Results of Roe’s scheme with source term added for Problem C

50

MacCormack Approach with hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-17a: Results of LeVeque & Yee’s MacCormack approach for Problem C

MacCormack Approach with hx = 0.001, ht = 0.0001 and t = 0 to 0.1.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

t)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 4-17b: Results of LeVeque & Yee’s MacCormack approach for Problem C

51

Comparison of the Different Flux-Limited Second Order Approaches at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Seabed MacCormack Roe's Scheme with Source Term Added Hubbard's Approach

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

0.35 0.4 0.45 0.5 0.55 0.6

Figure 4-18a: Comparison of the flux-limited second order results for Problem C

Comparison of the Different Flux-Limited Second Order Approaches at t = 0.1.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)


Figure 4-18b: Comparison of the flux-limited second order results for Problem C

52

Comparison of the Different Flux-Limited Second Order Approaches using a Larger Step-Size of hx = 0.01, ht = 0.001 and at t = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Riverbed MacCormack Roe's scheme with source term added Roe's scheme with source term decomposed

Figure 4-19a: Comparison of the flux-limited second order results for Problem C

Comparison of the Different Flux-Limited Second Order Approaches using a larger step size of hx = 0.01, ht = 0.001 and at t = 0.1.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

u(x,

0.1)

MacCormack Roe's scheme with source term added Hubbard's Approach

Figure 4-19b: Comparison of the flux-limited second order results for Problem C

53

Hubbard's Approach with hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Riverbed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-20a: Results of Hubbard’s approach for Problem D

Hubbard's Approach with hx = 1000, ht = 1 and t = 0 to 10800.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-20b: Results of Hubbard’s approach for Problem D

54

MacCormack Approach w ith hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Riverbed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-21a: Results of LeVeque & Yee’s MacCormack approach for Problem D

MacCormack Approach w ith hx = 1000, ht = 1 and t = 0 to 10800.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-21b: Results of LeVeque & Yee’s MacCormack approach for Problem D

55

Roe's Scheme w ith Source Term Added and hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Seabed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-22a: Results of Roe’s scheme with source term added for Problem D

Roe's Scheme with Source Term Added and hx = 1000, ht = 1 and t = 0 to 10800.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 4-22b: Results of Roe’s scheme with source term added for Problem D

56

Comparison of the Different Flux-Limited Second Order Approaches at t = 10800.

0

10

20

30

40

50

60

70

0 216000 432000 648000

x

Riverbed River InitiallyMacCormack Approach Roe's Scheme with Source Term AddedHubbard's Approach

Figure 4-23a: Comparison of the flux-limited second order results for Problem D

Comparison of the Different Flux-Lim ited Second Order Approaches at t = 10800.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 216000 432000 648000

x

u(x,

1080

0)


-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

216000 316000 416000 516000 616000

Figure 4-23b: Comparison of the flux-limited second order results for Problem D

57

5 Conclusion

In Chapter 4, it was shown that Roe’s scheme with source term decomposed and

Hubbard’s approach produced the most accurate numerical results for all four test

problems. All the other approaches were accurate for the first two test problems, but

as the source term became more significant, the other approaches became inaccurate.

In Problem D we saw that Roe’s scheme with source term decomposed and Hubbard’s

approach was the only approach that did not produce movement after x = 216,000m at

t = 10,800s. This is because the other approaches did not satisfy the C-property of

Bermudez & Vazquez[1].

Overall, Hubbard’s approach, which is Roe’s scheme with source term decomposed

with a flux-limiter applied to both the conservation law and the source term, produced

the most accurate numerical results. However, if we use the Superbee flux-limiter

instead of the Minmod flux-limiter with Hubbard’s approach then the approach

produces inaccurate numerical results. If we use (S-4) with Roe’s Superbee flux-

limiter, the Minmod flux-limiter and van Leer’s flux-limiter and use Δx = 1000, Δt = 1

and t = 0 to 10,800s, we may obtain Figure 5-1a to Figure 5-4b. Figure 5-1a to Figure

5-2b show that by using either Roe’s Superbee flux-limiter or van Leer’s flux-limiter

with Hubbard’s approach, oscillations occur in the numerical results. Figure 5-3a and

Figure 5-3b show that using the Minmod flux-limiter with Hubbard’s approach

produces no oscillations in the numerical results. Figure 5-4a and Figure 5-4b

compare the results obtained by using the three different flux-limiters with Hubbard’s

approach at t = 10,800s. Here, we can see that Roe’s Superbee flux-limiter has

58

produced the most oscillations, followed by van Leer’s flux-limiter and the Minmod

flux-limiter has produced no oscillations. This suggests that applying a flux-limiter to

the source term approximation when the source term is significant can create

oscillations in the numerical results. However, if we do not apply a flux-limiter to the

source term approximation but we do to the conservation law then we will no longer

be able to obtain a numerical scheme which satisfies the C-property of Bermudez &

Vazquez[1]. The only solution at present is to use the Minmod flux-limiter with

Hubbard’s approach or to use Roe’s first order scheme with source term decomposed.

Throughout this report, we have seen that adding a source term approximation can

produce accurate numerical results but as the source term becomes significant, adding

a source term approximation can give very inaccurate numerical results. We have

also shown that by decomposing the source term as well as the conservation law, we

may obtain very accurate numerical results for the first order case. However, when

applying flux-limiters to the source term as well as the conservation law to ensure the

numerical scheme satisfies the exact C-property, oscillations can occur in the

numerical solution depending on which flux-limiter is used.

59

Superbee Flux-Limiter with hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Seabed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 5-1a: Superbee Flux-Limiter results for Problem D

Superbe Flux-Limiter with hx = 1000, ht = 1 and t = 0 to 10800.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 5-1b: Superbee Flux-Limiter results for Problem D

60

van Leer Flux-Limiter with hx = 1000, ht = 1 and t = 0 to 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Seabed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 5-2a: van Leer Flux-Limiter results for Problem D

van Leer Flux-Limiter with hx = 1000, ht = 1 and t = 0 to 10800.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 5-2b: van Leer Flux-Limiter results for Problem D

61

Minmod Flux-Limiter with hx = 1000, ht = 1 and t = 0 t o 10800.

0

10

20

30

40

50

60

70

0 100000 200000 300000 400000 500000 600000

x

Seabed 0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 5-3a: Minmod Flux-Limiter results for Problem D

Minmod with hx = 1000, ht = 1 and t = 0 to 10800.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 100000 200000 300000 400000 500000 600000

x

u(x,

t)

0 1080 2160 3240 4320 5400 6480 7560 8640 9720 10800

Figure 5-3b: Minmod Flux-Limiter results for Problem D

62

Comparison of the Three Flux-Limiters at t = 10800.

0

10

20

30

40

50

60

70

0 216000 432000 648000

x

Riverbed River Initially Superbee van Leer Minmod

Figure 5-4a: Comparison of the Flux-Limiter results for Problem D

Comparison of the Three Flux-Limiters at t = 10800.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 216000 432000 648000

x

u(x,

1080

0)

Superbee van Leer Minmod

Figure 5-4b: Comparison of the Flux-Limiter results for Problem D

63

Acknowledgements

I would like to thank my supervisor Dr. P.K. Sweby and Dr. M.E. Hubbard for their

guidance and support. In addition, I would like to thank the EPSRC and HR

Wallingford for their funding.

64

References1. A. Bermudez and M^Elena Vazquez, Upwind Methods for Hyperbolic

Conservation Laws with Source Terms, Computers and Fluids Vol. 23, No. 8,1049 – 1071(1994).

2. P. Garcia-Navarro, Some Considerations and Improvements on the Performanceof Roe’s scheme for 1D Irregular Geometries, Internal Report 23 (1997).

3. P. Glaister, Difference Schemes for the Shallow Water Equations, NumericalAnalysis Report 9/87, University Of Reading (1987).

4. P. Glaister, Approximate Riemann Solutions of the Shallow Water Equations, J.Hydr. Research Vol. 26, No.3 (1988).

5. A. Harten, High Resolution Schemes for Conservation Laws, J. Comput. Phys. 49(1983).

6. M.E. Hubbard, Private Communication (University of Reading).

7. J. Hudson, Numerical Techniques for Conservation Laws with Source Terms, MScDissertation, University of Reading (1998).

8. D. Kroner, Numerical Schemes for Conservation Laws, Wiley-Teubner series(1997).

9. R.J. LeVeque and H.C. Yee, A Study of Numerical Methods for HyperbolicConservation Laws with Stiff Source Terms, J. Comput. Phys. 86, 187 – 210(1990).

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11. R.J. LeVeque, Balancing Source Terms and Flux Gradients in High-ResolutionGodunov Methods: The Quasi-Steady Wave-Propagation Algorithm, J. Comput.Phys. (1998).

12. P.L. Roe, Approximate Riemann Solvers, Parameter Vectors and DifferenceSchemes, J. Comput. Phys. 43, 357 – 372 (1981).

13. J.J Stoker, Water Waves, Wiley Interscience (1957).

14. P.K. Sweby, High Resolution Schemes Using Flux Limiters for HyperbolicConservation Laws, SIAM J. Num. Anal. 21, 995 (1984).

15. P.K. Sweby, Source Terms and Conservation Laws: A Preliminary Discussion,Numerical Analysis Report 6/89, University Of Reading (1989)

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